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Recently, the characterization of the compactness in the space BV([0, 1]) of functions of bounded Jordan variation was given. Here, certain generalizations of this result are given for the spaces of functions of bounded Waterman Μ-variation, Young Φ-variation and integral variation. It appears that, on the compact sets, the norm is uniformly approximated by certain seminorms induced by the selection of finitely many intervals in [0, 1]. [ABSTRACT FROM AUTHOR]

Purpose - The study hypothesized that the absence of the governing authority during the growth and expansion of informal settlements caused a highly randomized dense social fabric that shaped their characteristics. Design/methodology/approach - This research employs urban tactics and social theories to understand dynamic relationships in social consolidation in informal settlements in Jordan. The research adopted a mixedmethods approach, deploying qualitative and quantitative methods to understand the concepts, terms, perspectives, means and functions of open spaces in informal settlements. Findings - The results identified that the land ownership of open spaces, gender and age have a significant impact on the relationships and social interaction of people. The results suggested that despite the informal morphology of studied areas being random, unplanned and chaotic, there is often an underlying logic to meet occupants' needs. Research limitations/implications - This research explores informal spatiality to help understand the mechanisms of how marginal communities create and interact with each other in public spaces. This study is limited to the investigation of socio-cultural practices in public spaces, without an in-depth consideration of the roles of physical elements and features in the spatial configuration of these spaces. Originality/value - The importance of the research is that the exploration of informal spatiality of this neighborhood morphology will enable to understand the mechanism of how marginal communities create and interact among each other and their public spaces in different cities. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained. [ABSTRACT FROM AUTHOR]

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices A ∈ GL (d , R) , with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out that the scales of homogeneous and inhomogeneous Besov spaces differ in the way they depend on the dilation matrix: Two matrices A , B that induce the same scale of homogeneous Besov spaces also induce the same scale of inhomogeneous spaces, but the converse of this statement is generally false. We give a complete characterization of the different types of equivalence in terms of the Jordan normal forms of A , B. [ABSTRACT FROM AUTHOR]

We suggest a Hamiltonian formulation for the spin Ruijsenaars–Schneider system in the trigonometric case. Within this interpretation, the phase space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a representation space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh's formalism, we construct commuting Hamiltonian functions on the phase space and identify one of the flows with the spin Ruijsenaars–Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly. [ABSTRACT FROM AUTHOR]

In the first part of this paper we investigate four types of convergence of sequences of functions in metric spaces which preserve continuity. Besides a consideration of locally, quasi and continuously uniform convergence, we introduce the notion of semi uniform convergence. We discuss how all types of convergence are related both to each other and to pointwise convergence, and illustrate their behavior by examples. Moreover, we show how some of the types of convergence can be used to characterize compactness of the domains the functions under consideration live in. In the second part we investigate sequences of composition operators in the space BV of functions of bounded variation in the sense of Jordan. We give criteria under which such sequences converge locally uniformly and semi uniformly and present a new and short proof for the fact that composition operators which map the space BV into itself are automatically continuous. [ABSTRACT FROM AUTHOR]

In this paper we investigate the problem of uniform continuity of nonautonomous superposition operators acting between spaces of functions of bounded Λ-variation. In particular, we give the sufficient conditions for nonautonomous superposition operators to continuously map a space of functions of bounded Λ-variation into itself. The conditions cover the generators being functions of C1-class (in view of two variables), but also allow for less regular functions, including discontinuous generators. [ABSTRACT FROM AUTHOR]

We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers. However, each of the proposed concepts preserves only few properties of Jordan variation which are designed to a selected application. In contrast, the multivariate generalization of the Jordan space presented in this paper preserves all known and reveals some previously unknown properties of the space. These, in turn, are special cases of the basic properties of the introduced spaces proved in the paper. Specifically, the first part of the paper describes structure properties of functions of bounded (k , p) -variation (V p k functions). It includes assertions on discontinuity sets and pointwise differentiability of V p k functions and their Luzin type and C ∞ approximations. The second part presents results on Banach structure of V p k spaces, namely, atomic decomposition and constructive characterization of their predual spaces. As a result, we obtain the so-called two-stars theorems describing V p k spaces as second duals of their separable subspaces consisting of functions of "vanishing variation". [ABSTRACT FROM AUTHOR]